I would like to say that if the catode is not hot because it should not eject any electron through the tube and there's not any need for magnetic fields either to deflect or to accelerate the ray and there's no, supposedly, need for the flyback to polarize the screen to accelerate them in the last stage... the CRT should remain cold. However, I've come across CRTs which are hot even when there's no image on its screen. Is this normal or is a mal-functioning? Is it an error on the design? Thanks in advance. Regards,

I'm no expert but it doesn't sound that unusual to me. For one there's a heater element in the gun. And I don't think the image is completely off when there's no signal but it should be reduced power. Think of it this way, they need to be warmed up to get good color saturation and then there's the speed of presenting the image (i.e. when you turn a CRT on it's not instant perfect image).
I could see the same set being warmer than another like it if they have different brightness settings (i.e. the black level being less black on one of the sets).

Some CRT-based TVs keep a small current flowing through the filament so that the TV starts up faster. Therefore, the TV's power supply and other components are active even if the TV is "off". For many CRT TVs, "off" is really more of a "Standby" mode.